Soliton Scattering in Noncommutative Spaces

Abstract

We discuss exact multi-soliton solutions to integrable hierarchies on noncommutative space-times in diverse dimension. The solutions are represented by quasi-determinants in compact forms. We study soliton scattering processes in the asymptotic region where the configurations could be real-valued. We find that the asymptotic configurations in the soliton scatterings can be all the same as commutative ones, that is, the configuration of N-soliton solution has N isolated localized lump of energy and each solitary wave-packet lump preserves its shape and velocity in the scattering process. The phase shifts are also the same as commutative ones. As new results, we present multi-soliton solutions to noncommutative anti-self-dual Yang-Mills hierarchy and discuss 2-soliton scattering in detail.

New soliton solutions of anti-self-dual Yang-Mills equations

“…Asymptotic behaviors of these noncommutative soliton solutions were also proved to be the same as in commutative spaces [11,14]. It might be an interesting future work to confirm our conjecture that n soliton solutions in [11,12,14] have n isolated localized lumps of energy and preserve their shapes and velocities on each localized solitary wave lump. In addition, explicit analysis of these n soliton scatterings would be expected to give the phase shifts in the scattering processes as discussed in the standard soliton theory (e.g.…”

“…For multi-soliton solutions, we presented these discussions on noncommutative Euclidean spaces explicitly by the noncommutative Darboux transformation in [11] and the noncommutative Bäcklund transformation in [12,14]. Asymptotic behaviors of these noncommutative soliton solutions were also proved to be the same as in commutative spaces [11,14]. It might be an interesting future work to confirm our conjecture that n soliton solutions in [11,12,14] have n isolated localized lumps of energy and preserve their shapes and velocities on each localized solitary wave lump.…”

“…For multi-soliton solutions, we presented these discussions on noncommutative Euclidean spaces explicitly by the noncommutative Darboux transformation in [11] and the noncommutative Bäcklund transformation in [12,14]. Asymptotic behaviors of these noncommutative soliton solutions were also proved to be the same as in commutative spaces [11,14].…”

“…Another interesting problem is to compare the asymptotic behaviors of our solutions [11,12,14] with multi-soliton solutions in [9]. All of these studies might lead to a general formulation of Kodama's Grassmannian approach to the study of soliton scatterings [15], and give a new insight into Hirota's bilinear forms [25] or other formulation of integrable hierarchies [20,27].…”

Soliton solutions of noncommutative anti-self-dual Yang–Mills equations

“…The solutions include not only instantons but soliton-type solutions. Two soliton scatterings are discussed for G = GL(2) [21], where energy densities (the second Chern classes) are in general complex valued. For nsoliton scatterings, Wronskian-type solutions are suitable.…”